1. Spatial Indexing
SAMs

2. Spatial Indexing
Point Access Methods can index only points. What about regions?
Use the transformation technique
Z-ordering and quadtrees
New methods: Spatial Access Methods SAMs
R-tree and variations

3. Problem
Given a collection of geometric objects (points, lines, polygons, ...)
organize them on disk, to answer spatial queries (range, nn, etc)

4. R-tree z-ordering: cuts regions to pieces -> dup. elim.
how could we avoid that?
Idea: try to extend/merge B-trees and k-d trees
R-tree: a generalization of the B+-tree for multidimensional spaces

5. Kd-Btrees
[Robinson, 81]: if f is the fanout, split point-set in f parts; and so on, recursively

6. Kd-Btrees
But: insertions/deletions are tricky (splits may propagate downwards and upwards)
no guarantee on space utilization

7. R-trees [Guttman 84] Main idea: allow parents to overlap!
=> guaranteed 50% utilization
=> easier insertion/split algorithms.
(only deal with Minimum Bounding Rectangles - MBRs)

8. R-trees A multi-way external memory tree
Index nodes and data (leaf) nodes
All leaf nodes appear on the same level
Every node contains between m and M entries
The root node has at least 2 entries (children)

9. Example
eg., w/ fanout 4: group nearby rectangles to parent MBRs; each group -> disk page I C A G F H B J E D

10. Example
F=4 P1 P3 I C A G F H B J A B C H I J E P4 P2 D D E F G

11. Example F=4 P1 P3 I C A G F H B J E P4 P2 D P1 P2 P3 P4 A B C H I J D

12. R-trees - format of nodes
{(MBR; obj_ptr)} for leaf nodes P1 P2 P3 P4
x-low; x-high
y-low; y-high
...
obj
ptr A B C
...

13. R-trees - format of nodes
{(MBR; node_ptr)} for non-leaf nodes
x-low; x-high
y-low; y-high
...
node
ptr P1 P2 P3 P4
... A B C

14. R-trees:Search P1 P3 I C A G F H B J E P4 P2 D P1 P2 P3 P4 A B C H I J

15. R-trees:Search P1 P3 I C A G F H B J E P4 P2 D P1 P2 P3 P4 A B C H I J

16. R-trees:Search Main points:
every parent node completely covers its ‘children’
a child MBR may be covered by more than one parent - it is stored under ONLY ONE of them. (ie., no need for dup. elim.)
a point query may follow multiple branches.
everything works for any dimensionality

17. R-trees:Insertion Insert X P1 P3 I C A G F H B X J E P4 P2 D P1 P2 P3

18. R-trees:Insertion Insert Y P1 P3 I C A G F H B J Y E P4 P2 D P1 P2 P3

19. R-trees:Insertion Extend the parent MBR P1 P3 I C A G F H B J Y E P4

20. R-trees:Insertion How to find the next node to insert the new object?
Using ChooseLeaf: Find the entry that needs the least enlargement to include Y. Resolve ties using the area (smallest)
Other methods (later)

21. R-trees:Insertion P1 K P3 I C A G W F H B J E P4 P2 D
If node is full then Split : ex. Insert w P1 K P3 I P1 P2 P3 P4 C A G W F H B J A B C K H I J E P4 P2 D D E F G

22. R-trees:Split (A1: plane sweep, until 50% of rectangles) P1 K
Split node P1: partition the MBRs into two groups.
(A1: plane sweep,
until 50% of rectangles)
A2: ‘linear’ split
A3: quadratic split
A4: exponential split:
2M-1 choices P1 K C A W B

23. R-trees:Split seed2 R seed1 pick two rectangles as ‘seeds’;
assign each rectangle ‘R’ to the ‘closest’ ‘seed’
seed2 R
seed1

24. R-trees:Split seed2 R seed1 pick two rectangles as ‘seeds’;
assign each rectangle ‘R’ to the ‘closest’ ‘seed’:
‘closest’: the smallest increase in area
seed2 R
seed1

25. R-trees:Split How to pick Seeds:
Linear:Find the highest and lowest side in each dimension, normalize the separations, choose the pair with the greatest normalized separation
Quadratic: For each pair E1 and E2, calculate the rectangle J=MBR(E1, E2) and d= J-E1-E2. Choose the pair with the largest d

26. R-trees:Insertion
Use the ChooseLeaf to find the leaf node to insert an entry E
If leaf node is full, then Split, otherwise insert there
Propagate the split upwards, if necessary
Adjust parent nodes

27. R-Trees:Deletion Other method (later)
Find the leaf node that contains the entry E
Remove E from this node
If underflow:
Eliminate the node by removing the node entries and the parent entry
Reinsert the orphaned (other entries) into the tree using Insert
Other method (later)

28. R-trees: Variations
R+-tree: DO not allow overlapping, so split the objects (similar to z-values)
R*-tree: change the insertion, deletion algorithms (minimize not only area but also perimeter, forced re-insertion )
Hilbert R-tree: use the Hilbert values to insert objects into the tree